I have attempted to show that infinitely many primes are in an arithmetic progression of length 3 in the primes following Ben Green's comment that one can do this using the circle method; but I have not found any success. Can anyone suggest (with more detail perhaps) a way to show that infinitely many primes are in an arithmetic progression of length at least 3?

Edit: In view of the comments, I have rephrased the question: My intention was to ask what can one do (Ben Green suggested the circle method, but gave no details) to show that 'most' primes (that is, the exceptional set has upper density 0) are in an arithmetic progression of at least 3 in the primes.

The title question is not the same as the question you give in the body. The answer to the title question is "yes" by the Green-Tao theorem. You can find an answer to the body question in a 1939 paper of van der Corput and a 1944 paper of Chowla (see the references in Green-Tao).
–
S. Carnahan♦Mar 4 '11 at 17:55

4

Instead of "infinitely many primes in an AP of length 3", you mean "infinitely many APs of length 3 in the primes".
–
Kevin O'BryantMar 4 '11 at 18:17

1

I am afraid the question still does not actually ask what I believe you want to ask. My original reading of the question was that you want suggestions on how to do what Ben Green suggested was doable (but, likely, not written anywhere). If you actually just want 'infinitely many' and not 'almost all' then as said by others the answer is known; and if you want a good reference for this result, I think it is simplest to explictly ask for this. Also, some confusion here could be avoided by suplementing the verbal description with a formula.
–
quidMar 4 '11 at 19:58

1

Cont. While I am not competent to judge this, it might be the case that you can get an impression what Ben Green had in mind by looking at his paper 'Roth's theorem in the primes'; there is also a more recent paper by Helfgott and de Roton 'Improving Roth's theorem in the primes.'
–
quidMar 4 '11 at 20:10

The question is about what can be proved, and I am in no way qualified to comment on that. But just to be clear, there is probably a big gap between what can be proved and what it would be reasonable to suspect is true. I'd imagine that every odd prime is the smallest member of infinitely many 3 term prime APs and even that for all $\epsilon \gt 0$ there are only finitely many primes not in an AP with largest member less than $(1+\epsilon)p$ or even largest member less than $p^{1+\epsilon}$.
–
Aaron MeyerowitzMar 7 '11 at 0:01

1 Answer
1

It is a theorem of Ben Green that every subset of the primes of positive relative density contains a progression of length three. As an immediate consequence, the set of primes $A$ which are not the first term in a progression of primes of length three has density zero (otherwise $A$ would contain a length three progression, a contradiction).

Ben's proof is, strictly speaking, an application of the circle method, but is probably overkill for your problem (roughly speaking, Ben wants to find a length three progression with all three elements in an arbitrary dense subset $A$ of the primes; for your problem, one only needs to study the simpler problem where the smallest term of the progression needs to be in $A$ but the other two elements lie in the set $P$ of primes). So a simpler proof (still by the circle method) is likely to exist. (Note, by the way, that a later paper of Ben and myself gives a slightly simpler proof of Ben's theorem (and, of course, we also have a more complicated proof as well).)

Another approach is to modify the argument of Montgomery and Vaughan (also, ultimately, based on the circle method) that shows that the number of even numbers that are not the sum $p_1+p_2$ of two primes is very low in density (much lower than the density of the primes, in particular). (Actually, an older and somewhat simpler paper of Vaughan already suffices for this.) The same argument should also show that the odd integers $n$ that are not the first term $2p_1-p_2$ of an 3-term AP whose other two terms $p_1,p_2$ are primes larger than $n$, also has density much smaller than that of the primes.